1. Scaling, renormalization and self-similarity in complex networks
Hernan A. Makse
Levich Institute and Physics Dept.
City College of New York
Chaoming Song (CCNY)
Lazaros Gallos (CCNY)
Shlomo Havlin (Bar-Ilan, Israel)
Protein interaction network

2. Are “scale-free” networks really ‘free-of-scale’?
“If you had asked me yesterday, I would have said surely not” - said Barabasi. (Science News, February 2, 2005).
Small World effect shows that distance between nodes grows logarithmically with N (the network size): OR
Self-similar = fractal topology is defined by a power-law relation:
Small world contradicts self-similarity!!!
How the network behaves under a scale transformation.

3. WWW
nd.edu
R. Albert, et al., Nature (1999)
300,000 web-pages

4. k
P(k)
Internet
Faloutsos et al., SIGCOMM ’99
Internet connectivity, with selected backbone ISPs (Internet Service Provider) colored separately.

5. from MIPS database, mips.gsf.de
Yeast Protein-Protein Interaction Map
Individual proteins
Physical interactions from the “filtered yeast interactome” database: 2493 high-confidence interactions observed by at least two methods (yeast two-hybrid).
1379 proteins, <k> = 3.6
J. Han et al., Nature (2004)
Modular structure
according to function!
Colored according to protein function in the cell:
Transcription, Translation, Transcription control, Protein-fate, Genome maintenance, Metabolism, Unknown, etc
from MIPS database, mips.gsf.de

6. Metabolic network of biochemical reactions
in E.coli
Chemical substrates
Biochemical interactions:
enzyme-catalyzed reactions that
transform one metabolite into another.
J. Jeong, et al., Nature, (2000)
Modular structure
according to the biochemical
class of the metabolic products
of the organism.
Colored according to product class:
Lipids, essential elements, protein, peptides and amino acids, coenzymes and prosthetic groups, carbohydrates, nucleotides and nucleic acids.

7. How long is the coastline of Norway?
It depends on the length of your ruler.
Fractals look the same on all scales = `scale-invariant’.
Box length
Fractal Dimension dB-
Box Covering Method
Total no. of boxes

8. We need the minimum number of boxes: NP-complete optimization problem!
Boxing in Biology
Boxing in Biology
How to “zoom out” of a complex network?
Generate boxes where all
nodes are within a distance
Calculate number of boxes, ,
of size needed to cover the
network
We need the minimum number of boxes: NP-complete optimization problem!

9. Most efficient tiling of the network
8 node network: Easy to solve
4 boxes
5 boxes
300,000 node network: Mapping to graph colouring problem. Greedy algorithm to find minimum boxes 1 2

10. Larger distances need fewer boxes1 2
-dB
fractal
log(NB) 3
non fractal
log(lB)

11. Box covering in yeast: protein interaction network

12. Most complex networks are Fractal
Biological networks
Metabolic
Protein interaction
43 organisms
- all scale
Three domains of life:
archaea, bacteria, eukaria
E. coli, H. sapiens, yeast
Song, Havlin, Makse, Nature (2005)

13. Metabolic networks are fractals

14. Technological and Social Networks TOO
WWW
Hollywood film actors
212,000 actors
Other bio networks: Khang and Bremen groups
Internet is not fractal!
nd.edu domain
300,000 web-pages

15. Cluster growing method
Two ways to calculate fractal dimensions
Box covering method
Cluster growing method
In homogeneous systems (all nodes with similar k)
both definitions agree:
percolation

16. Box Covering= flat average Cluster Growing = biased
exponential
power law
Different methods yield different results due to heterogeneous topology
Box covering reveals the self similarity. Cluster growth reveals the small world.
NO CONTRADICTION! SAME HUBS ARE USED MANY TIMES IN CG.

17. Is evolution of the yeast fractal?
present
day
Archaea
+ Bacteria
Animals
+ Plants
Other
Fungi
Yeast
~ 300 million
years ago
Ancestral yeast
Ancestral Fungus
Ancestral Eukaryote
1 billion
years ago
Ancestral Prokaryote Cell
3.5 billion years ago
Following the phylogenetic
tree of life:
COG database
von Mering, et al Nature (2002)
1.5 billion years ago

18. Same fractal dimension and scale-free exponent over 3.5 billion years…
Suggests that present-day networks could have
been created following a self-similar, fractal dynamics.

19. Renormalization in Complex Networks
NOW, REGARD
EACH BOX AS
A SINGLE NODE
AND ASK WHAT
IS THE DEGREE
DISRIBUTION OF
THE NETWORK
OF BOXES AT
DIFFERENT
SCALES ?

20. Renormalization of WWW network with

21. The degree distribution is invariant under renormalization
Internet is not fractal dB--> infinity
But it is renormalizable

22. Turning back the time
Repeatedly BOXING the network is the same as going back in time:
from a single node to present day.
THE RENORMALIZATION SCHEME
renormalization
present day
network
ancestral
node 1
time evolution
Can we “predict” the past…. ? if not the future.

23. Evolution of complex networks
opening boxes
time evolution

24. How does Modularity arise?
The boxes have a physical meaning =
self-similar nested communities
How to identify communities in complex networks?
renormalization
present day
network
ancestral
node 1
time evolution

25. Emergence of Modularity in PIN
Boxes are related to the biologically relevant functional modules
in the yeast protein interactome
renormalization
time evolution
translation
transcription
protein-fate
cellular-fate
organization
present day
network
ancestral
cell

26. Emergence of modularity in metabolic networks
Appearance of functional modules in E. coli metabolic network.
Most robust network than non-fractals.

27. How the communities/modules are linked?
Theoretical approach
How the communities/modules are linked?
renormalization
k’=2
k=8
s=1/4
k: degree of the
nodes
k’: degree of the
communities
node degree
community
degree
factor<1

28. Theoretical approach to modular networks: Scaling theory to the rescue
WWW
The larger the community
the smaller their connectivity
new exponent describing how
families link

29. A theoretical prediction relating the different exponents
Scaling relations
A theoretical prediction relating the different exponents
distance
new exponent
boxes
degree
new scaling relation

30. The communities also follow a self-similar pattern
Scaling relations
The communities also follow a self-similar pattern
WWW
Metabolic
prediction
Scaling relation
works
scale-free
fractals
communities/modules

31. Why fractality? Some real networks are not fractal
INTERNET
Other models fail too: Erdos-Renyi, hierarchical model, fitness model, JKK model, pseudo-fractals models, etc.
The Barabasi-Albert model of preferential
attachment does not generate fractal networks
All the models fail to predict self-similarity

32. What is the origin of self-similarity?
Can you see the difference?
FRACTAL
NON FRACTAL
Internet map
E.coli metabolic map
Yeast protein map
HINT: the key to understand fractals is in the degree
correlations P(k1,k2) not in P(k)

33. Quantifying correlations
P(k1,k2):
Probability to find a node with k1 links connected with a node of k2 links
Internet map - non fractal
Metabolic map - fractal
high prob.
low prob.
log(k2)
log(k2)
P(k1,k2)
low prob.
high prob.
log(k1)
log(k1)
Hubs connected with hubs
Hubs connected with non-hubs

34. Quantify anticorrelation between hubs at all length scales
Hub-Hub Correlation function: fraction of hub-hub connections
hubs
Renormalize
hubs
Hubs connected directly

35. Hub-hub connection organized in a self-similar way
non-fractal
The larger de implies more anticorrelation
fractal
(fractal) (non-fractal)
Anticorrelations are essential for fractal structure

36. What is the origin of self-similarity?
Non-fractal networks
Fractal networks
very compact networks
hubs connected with other hubs
strong hub-hub “attraction”
assortativity
less compact networks
hubs connected with non-hubs
strong hub-hub “repulsion”
dissasortativity
Internet
All available models: BA model, hierarchical
random scale free, JKK, etc
WWW, PIN, metabolic, genetic, neural
networks, some sociological networks

37. How to model it? renormalization reverses time evolution
Song, Havlin, Makse, Nature Physics, 2006
Both mass and degree increase exponentially with time
time
renormalize
offspring nodes attached to their parents
(m=2) in this case
Scale-free:
Mode I
Mode II

38. How does the length increase with time?
Mode I: NONFRACTAL
SMALL WORLD
Mode II: FRACTAL

39. Combine two modes together
Mode I with probability e Mode II with probability 1-e
time
renormalize
e=0.5

40. Predictions
Model reproduces local small world, scale-free and fractality
yeast
h.sapiens
model with e=0.2
repulsion between hubs leads to fractal topology
small world locally inside well defined communities
model with e=1
attraction between hubs
non-fractal
small world globally

41. The model reproduces the main features of real networks
Case 1: e = 0.8: FRACTALS Case 2: e = 1.0: NON-FRACTALS

42. Model predicts all exponents in terms of growth rates
Each step the total mass scales with a constant n, all the degrees scale with a constant s.
The length scales with a constant a, we obtain:
We predict the fractal exponents:

43. Time evolution in yeast network

44. Multiplicative and exponential growth in yeast PIN
Length-scales, number of conserved proteins and degree

45. at the expense of the “poor”=
A new principle of network dynamics
1930
solid-state physics
big world
1960
Erdos-Renyi model
small world
democracy=
socialism
1999
BA model
“rich-get-richer”=
capitalism
2005
fractal model
“rich-get-richer”
at the expense of the “poor”=
globalization
less vulnerable to intentional attacks

46. Positions available: jamlab.org
Summary
In contrast to common belief, many real world networks are self-similar.
FRACTALS: WWW, Protein interactions, metabolic networks, neural networks, collaboration networks.
NON-FRACTALS: Internet, all models.
Communities/modules are self-similar, as well.
Scaling theory describes the dynamical evolution.
Boxes are related to the functional modules in metabolic and protein networks.
Origin of self similarity: anticorrelation between hubs
Fractal networks are less vulnerable than non-fractal networks
Positions available: jamlab.org

47. An finally, a model to put all this together
A multiplicative growth process
of the number of nodes and links
m = 2
Analogous to
duplication/divergence
mechanism in proteins??
Probability e
hubs always connected
strong hub attraction
should lead to non-fractal
Probability 1-e
hubs never connected
strong hub repulsion
should lead to fractal

48. Different growth modes lead to different
topologies
For the both models, each step the total number of nodes scale as n = 2m +1( N(t+1) = nN(t) ). Now we investigate the transformation of the lengths. They show quite different ways for this two models as following:
Mode I: L(t+1) = L(t)+2
Then we lead to two different scaling law of N ~ L
smaller
smaller
Mode II: L(t+1) = 2L(t)+1
Mode III: L(t+1) =3L(t)

49. Dynamical model
Suppose we have e probability to have mode I, 1-e probability to have mode II and mode III. Then we have: or

50. Graph theoretical representation of a metabolic
network
(a) A pathway (catalyzed by Mg2+-dependant enzymes).
(b) All interacting metabolites are considered equally.
(c) For many biological applications it is useful to ignore co-factors, such as the high energy-phosphate donor ATP, which results in a second type of mapping that connects only the main source metabolites to the main products.

51. Classes of genes in the yeast proteome

52. Renormalization following the phylogenetic tree
P. Uetz, et al. Nature 403 (2000).